Edexcel Maths C3 Topic Questions from Papers Differentiation casperyc.weebly.com
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Edexcel Maths C3
Topic Questions from Papers
Differentiation
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4. f(x) = 3ex – ln x – 2, x > 0.
(a) Differentiate to find f ´(x).(3)
The curve with equation y = f(x) has a turning point at P. The x-coordinate of P is α.
(b) Show that α = e–α.
(2)
The iterative formula
is used to find an approximate value for α.
(c) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.
(2)
(d) By considering the change of sign of f ´(x) in a suitable interval, prove that α = 0.1443
correct to 4 decimal places.
(2)
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11 06
e , 1,nxnx x−+ = =
16
12
C3EdexcelJune05QP
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P x P
P y ax ba b
y x⎛ ⎞= ⎜ ⎟⎝ ⎠
C3EdexcelJan06QP
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x
x x
x y xyx
xx
C3EdexcelJan06QP
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2. Differentiate, with respect to x,
(a) e3x + ln 2x,(3)
(b) .(3)
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322(5 )x
Q2
(Total 6 marks)
C3EdexcelJune06QP
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3. The curve C has equation
x = 2 sin y.
(a) Show that the point lies on C.(1)
(b) Show that at P.(4)
(c) Find an equation of the normal to C at P. Give your answer in the form y =mx + c,where m and c are exact constants.
(4)
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d 1dyx
,4
P
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4. (i) The curve C has equation
Use calculus to find the coordinates of the turning points of C.(6)
(ii) Given that
find the value of .(5)
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d 1at ln 3d 2y xx
32 2(1 e ) ,xy
2 .9xyx
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Cy x x
yx
C
yx
C
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C
y x x nxe2 2 12
tan , ( )
C tan 1x
C 0x
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C
y x x3 2 4 2sin cos , x
A C
C A
y R xsin( ),2 R 02
C x
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P
y P
x P.
P y ax ba b
y e x= +4 2 1.
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x
x
x x x
y x xx
x
yx x
yx
yx
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x
y x x
x
yx
xx
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y ax b a b
x y
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y x
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C
2)35(3x
y−
=35≠x
P C x C P0,ax by c+ + = a b c
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C 2(2 5 2)e .xy x x −= − +
C y
C x 2x = xC x
x
y
dd
C
O
y
x
C
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7. The curve C has equation
3 sin 22 cos 2
xyx
+=+
(a) Show that
( )2d 6sin 2 4cos 2 2d 2 cos 2y x xx x
+ +=+
(4)
(b) Find an equation of the tangent to C at the point on C where x = π2
. Write your answer in the form y = ax + b, where a and b are exact constants.
(4)
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C3EdexcelJan11QP
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8. (a) Given that
( )d cos sind
x xx
= −
show that ( )d sec sec tan .d
x x xx
=
(3)
Given that
sec 2x y=
(b) find dd
xy
in terms of y.
(2)
(c) Hence find ddyx
in terms of x.
(4)
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1. Differentiate with respect to x
(a) ln x x2 3 5+ +( )(2)
(b) cos x
x2(3)
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C3EdexcelJune11QP
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7. f ( )( )( )
, ,x xx x
xx
x x= −+ −
−−
≠ ± ≠ −4 52 1 3
29
3 122
(a) Show that f ( )
( )( )x
x x=
+ +5
2 1 3(5)
The curve C has equation y = f (x). The point P − −⎛⎝⎜
⎞⎠⎟
1 52
, lies on C.
(b) Find an equation of the normal to C at P.(8)
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C3EdexcelJune11QP
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8. (a) Express 2cos 3x – 3sin 3x in the form R cos (3x + ), where R and are constants, R 0 and 0
2< <
πα . Give your answers to 3 significant figures.(4)
f e( ) cosx xx= 2 3
(b) Show that f ′(x) can be written in the form
′ =f ( ) e cosx R x2 +( )x3 α
where R and are the constants found in part (a).(5)
(c) Hence, or otherwise, find the smallest positive value of x for which the curve with equation y = f (x) has a turning point.
(3)
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C3EdexcelJune11QP
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1. Differentiate with respect to x, giving your answer in its simplest form,
(a) x x2 3ln ( )(4)
(b) sin 43
x
x (5)
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C3EdexcelJan12QP
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4. The point P is the point on the curve x y= +⎛⎝⎜
⎞⎠⎟
212
tan π
with y-coordinate
4π .
Find an equation of the normal to the curve at P.
(7)
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3.
Figure 1
Figure 1 shows a sketch of the curve C which has equation
y xx= e 3 3sin , − 3 3� �xπ π
(a) Find the x coordinate of the turning point P on C, for which x� 0 Give your answer as a multiple of �.
(6)
(b) Find an equation of the normal to C at the point where x = 0(3)
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O
PC
y
x
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7. (a) Differentiate with respect to x,
(i) x x12 3ln( )
(ii) 1 102 1 5
−−
xx( )
, giving your answer in its simplest form.(6)
(b) Given that x y= 3 2tan find ddyx
in terms of x. (5)
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C3EdexcelJune12QP
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1. The curve C has equation
y x= −( )2 3 5
The point P lies on C and has coordinates (w, – 32).
Find
(a) the value of w,(2)
(b) the equation of the tangent to C at the point P in the form y mx c= + , where m and c are constants.
(5)
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C3EdexcelJan13QP
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5. (i) Differentiate with respect to x
(a) y x x= 3 2ln
(b) y x x= +( sin )2 3
(6)
Given that x y= cot ,
(ii) show that ddyx x
= −+
11 2
(5)
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C3EdexcelJan13QP
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5. Given that
x = sec2 3y, 0 < y < 6π
(a) find dd
xy
in terms of y.(2)
(b) Hence show that
ddyx x x
=−
1
6 112( )
(4)
(c) Find an expression for dd
2
2y
x in terms of x. Give your answer in its simplest form.
(4)
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